Algebraic Number Fields (eBook)

Front Cover 1
Algebraic Number Fields 4
Copyright Page 5
Contents 8
Preface 10
Chapter I. Subrings of Fields 12
1. Localization 12
2. Integral Dependence 15
3. Discrete Valuation Rings and Dedekind Rings 19
4. Fractional Ideals and the Class Group 27
5. Norms and Traces 29
6. Extensions of Dedekind Rings 34
7. Discriminant 40
8. Norms of Ideals 46
9. Cyclotomic Fields 50
10. Lattices in Real Vector Spaces 59
11. The Unit Theorem and Finiteness of the Class Number 63
Chapter II. Complete Fields 77
1. Valuations 77
2. Completions 83
3. Extensions of Nonarchimedean Valuations 89
4. Archimedean Valuations 96
5. Local Norms and Traces and the Product Formula 99
Chapter III. Decomposition Groups and the Artin Map 105
1. Decomposition and Inertia Groups 105
2. The Frobenius Automorphism 108
3. The Artin Map for Abelian Extensions 113
Chapter IV. Analytic Methods 117
1. Moduli and Ray Classes 118
2. Dirichlet Series 124
3. Characters of Abelian Groups 137
4. L-Series and Product Representations 138
5. Frobenius Density Theorem 144
Chapter V. Class Field Theory 151
1. Cohomology of Cyclic Groups 151
2. Preparations for the Second Inequality 154
3. A Norm Index Computation 158
4. The Fundamental Equality for Cyclic Extensions 165
5. The Reciprocity Theorem 168
6. Ideal Groups, Conductors, and Class Fields 177
7. Reduction Steps Toward the Existence Theorem 180
8. Kummer Extensions and The S-Unit Theorem 182
9. The Existence Theorem 184
10. Some Consequences of the Classification Theorem 191
11. Preliminaries for the Norm Residue Map and the Conductor Theorem 194
12. Norm Residue Symbol 197
13. The Hilbert Class Field 202
Chapter VI. Application of the General Theory to Quadratic Fields 207
1. The Conductor of Q(vd) 207
2. Two Examples 209
3. The Extended Class Group 214
Appendix A. Normal Basis Theorem and Hilbert’s Theorem 90 223
Appendix B. Modules over Principal Ideal Domains 225
Bibliography 229
Index 230